If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?
The density matrix describes the state of the system, so its exact form will depend on the state of the system.
If H1 and H2 correspond to different particles, then the density matrix will be block diagonal only if the particles are not entangled (corresponding to a product state).